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Characterization of the logistic and loglogistic distributions by extreme value related stability with random sample size

Published online by Cambridge University Press:  14 July 2016

W. J. Voorn
W. J. Voorn*
Affiliation:
Universiteit van Amsterdam
*
∗Postal address: Universiteit van Amsterdam, Vakgroep Medische Fysica, Meibergdreef 15, 1105 AZ Amsterdam, The Netherlands.
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Abstract

Maximum stability of a distribution with respect to a positive integer random variable N is defined by the property that the type of distribution is not changed when considering the maximum value of N independent observations. The logistic distribution is proved to be the only symmetric distribution which is maximum stable with respect to each member of a sequence of positive integer random variables assuming value 1 with probability tending to 1. If a distribution is maximum stable with respect to such a sequence and minimum stable with respect to another, then it must be logistic, loglogistic or ‘backward' loglogistic. The only possible sample size distributions in these cases are geometric.

Keywords

EXTENDED LOGISTIC DISTRIBUTIONEXTENDED LOGLOGISTIC DISTRIBUTIONGEOMETRIC DISTRIBUTIONEXTENDED GEOMETRIC DISTRIBUTIONSYMMETRIC DISTRIBUTIONSINCREASING FAILURE RATE
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 4 , December 1987 , pp. 838 - 851
Copyright
Copyright © Applied Probability Trust 1987 

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